SPEAKERS
Jean Bourgain, Institute for Advanced Study, Luis Caffarelli, University of Texas, Austin Panagiota Daskalopoulos, Columbia University, New York Donatella Danielli, Purdue University, West Lafayette Guy David, Université de Paris Sud Orsay David Jerison, Massachusets Institute of Technology, Cambridge Frank Merle, Université de Cergy-Pontoise Irina Mitrea, Cornell University, Ithaca Alberto Ruiz, Universidad Autónoma de Madrid Zhongwei Shen, University of Kentucky, Lexington Gigliola Staffilani, Massachusets Institute of Technology, Cambridge Elias Stein, Princeton University Luis Vega, Universidad del País Vasco, Bilbao Gregory Verchota, Syracuse University
TITLES AND ABSTRACTS
Sum-product and exponential sum estimates Jean Bourgain To be announced Luis Caffarelli Vanishing behavior of maximal solutions to the Ricci flow on R2 Panagiota Daskalopoulos To be announced Donatella Danielli Quasiminimal sets for Hausdorff measure Guy David Abstract: These sets were also called restricted sets by F. Almgren. We shall explain why they may come up when trying to minimize a functional like \int E h(x) dHk(x) under geometrical constraints. If h is merely bounded from above and below, not much more information may be available, and also quasiminimality is enough to get nontrivial information. The regularity and speed of the Hele-Shaw flow from an initial Lipschitz surface David Jerison Abstract: I will talk about joint work with Sunhi Choi and Inwon Kim in which we prove that for small positive time, the solution to the one-phase Hele-Shaw flow, starting from an initial surface with small Lipschitz constant, is smooth. Along the way we obtain upper and lower bounds on the speed of the free boundary in terms of the initial data. Review on blow-up phenomenon for critical NLS Frank Merle Abstract: We will give recent results on the qualitative properties of blow-up solution for crtial NLS and related problems. Global Optimization Techniques for Singular Integrals Irina Mitrea Abstract: We survey recent progress in the direction of understanding the spectra of integral operators which arise naturally in the context of elliptic boundary problems in non-smooth domains. The focus is the Spectral Radius Conjecture, for which we present both positive and negative results, some of which have been obtained via computer aided proofs. Partial recovery of a potential from backscattering Alberto Ruiz Lp Estimates for Elliptic Systems on Lipschitz Domains Zhongwei Shen Abstract: In this talk we discuss a new approach to the Lp Dirichlet problem via L2 estimates and a real variable argument. The approach may be applied to elliptic systems as well as higher order elliptic equations on Lipschitz domains in higher dimensions. Related work on the Lp boundedness of the Riesz transforms associated with second order elliptic equations will also be discussed. Symplectic non-squeezing of the KdV flow Gigliola Staffilani Abstract: In this talk I will present a Gromov's symplectic non-squeezing result for the KdV flow on the torus, obtained in collaboration with J. Colliander, M. Keel, H. takaoka and T. Tao. A similar result was obtained by Bourgain for the cubic NLS flow. Our theorem is on the line of work of Kuksin who initiated the investigation into non-squeezing results for infinite dimensional Hamiltonian systems. Our approach follows the one proposed by Bourgain . This approach is based on projecting the infinite dimensional symplectic phase space into the first N modes, using here Gromov's non-squeezing theorem and then passing to the limit. In the KdV contest a major difficulty is the lack of any sort of smoothing estimate which would allow us to easily approximate the infinite dimensional KdV flow by the finite dimensional projected Hamiltonian flow, as Bourgain did. To resolve the problem we are forced to invert the Miura transform and work on the level of the modified KdV equation, for which smoother estimates are available. Some discrete analogues on the Heisenberg group Elias Stein The initial value problem for quasi-linear Schrödinger equations Luis Vega To be announced Gregory Verchota